Large sparse linear systems arising from mimetic discretization

AbstractIn this work we perform an experimental study of iterative methods for solving large sparse linear systems arising from a second-order 2D mimetic discretization. The model problem is the 2D Poisson equation with different boundary conditions. We use GMRES with the restarted parameter and BiCGstab as iterative methods. We also use various preconditioning techniques including the robust preconditioner ILUt. The numerical experiments consist of large sparse linear systems with up to 643200 degrees of freedom

Hybrid mimetic finite-difference and virtual element formulation for coupled poromechanics

We present a hybrid mimetic finite-difference and virtual element formulation for coupled single-phase poromechanics on unstructured meshes. The key advantage of the scheme is that it is convergent on complex meshes containing highly distorted cells with arbitrary shapes. We use a local pressure-jump stabilization method based on unstructured macro-elements to prevent the development of spurious pressure modes in incompressible problems approaching undrained conditions. A scalable linear solution strategy is obtained using a block-triangular preconditioner designed specifically for the saddle-point systems arising from the proposed discretization. The accuracy and efficiency of our approach are demonstrated numerically on two-dimensional benchmark problems.Comment: 25 pages, 17 figure

Parallel solvers for virtual element discretizations of elliptic equations in mixed form

The aim of this paper is twofold. On the one hand, we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three-dimensional elliptic equations on polyhedral meshes. On the other hand, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. Both these topics are numerically validated by several parallel tests performed on a Linux cluster. More specifically, we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3,700 are used to verify the accuracy and physical fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational Physic

An algebraic multigrid method for Q2−Q1Q_2-Q_1 mixed discretizations of the Navier-Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Specifically, we investigate a $Q_2-Q_1$ mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches leveraging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the $Q_2-Q_1$ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the finest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa

A performance analysis of a mimetic finite difference scheme for acoustic wave propagation on GPU platforms

Realistic applications of numerical modeling of acoustic wave dynamics usually demand high-performance computing because of the large size of study domains and demanding accuracy requirements on simulation results. Forward modeling of seismic motion on a given subsurface geological structure is by itself a good example of such applications, and when used as a component of seismic inversion tools or as a guide for the design of seismic surveys, its computational cost increases enormously. In the case of finite difference methods (or any other volumen-discretization scheme), memory and computing demands rise with grid refinement, which may be necessary to reduce errors on numerical wave patterns and better capture target physical devices. In this work, we present several implementations of a mimetic finite difference method for the simulation of acoustic wave propagation on highly dense staggered grids. These implementations evolve as different optimization strategies are employed starting from appropriate setting of compilation flags, code vectorization by using streaming SIMD extensions Advanced Vector Extensions (AVX), CPU parallelization by exploiting the Open Multi-Processing framework to the final code parallelization on graphics processing unit platforms. We present and discuss the increasing processing speed up of this mimetic scheme achieved by the gradual implementation and testing of all these performance optimizations. In terms of simulation times, the performance of our graphics processing unit parallel implementations is consistently better than the best CPU version.Authors from Universidad Central de Venezuela (UCV) were partially supported by: Consejo de Desarrollo Científico y Humanístico de la UCV, Vice-rectorado Académico de la UCV, Coordinación de Investigación de la Facultad de Ciencias UCV, Banco Central de Venezuela (BCV) and Generalitat Valenciana under project PROMETEOII/2015/015